**Purpose**:
To ascertain the theoretical impact of anatomical variations in the effective lens position (ELP) of the intraocular lens (IOL) in a thick lens eye model. The impact of optimization of IOL power formulas based on a single lens constant was also simulated.

**Methods**:
A schematic eye model was designed and manipulated to reflect changes in the ELP while keeping the optical design of the IOL unchanged. Corresponding relationships among variations in ELP, postoperative spherical equivalent refraction, and required IOL power adjustment to attain target refractions were computed for differing corneal powers (38 diopters [D], 43 D, and 48 D) with IOL power ranging from 1 to 35 D.

**Results**:
The change in ELP required to compensate for various systematic biases increased dramatically with low-power IOLs (less than 10 D) and was proportional to the magnitude of the change in refraction. The theoretical impact of the variation in ELP on postoperative refraction was nonlinear and highly dependent on the optical power of the IOL. The concomitant variations in IOL power and refraction at the spectacle plane, induced by varying the ELP, were linearly related. The influence of the corneal power was minimal.

**Conclusions**:
The consequences of variations in the lens constant mainly concern eyes receiving high-power IOLs. The compensation of a systematic bias by a constant increment of the ELP may induce a nonsystematic modification of the predicted IOL power, according to the biometric characteristics of the eyes studied.

**Translational Relevance**:
Optimizing IOL power formulas by altering the ELP may induce nonsystematic modification of the predicted IOL power.

^{1}The formula precision and accuracy depend on the reliability of the preoperative biometric measurements, the difference between the achieved versus predicted effective lens position (ELP), and possible postoperative fluctuations in corneal power.

^{2}

^{–}

^{4}The quality of the postoperative ELP prediction is the most critical factor in controlling residual refractive error.

^{5}The ELP does not necessarily coincide with the physical position of the IOL front surface, equatorial plane, or back surface. In theoretical thick lens paraxial ocular models, the ELP corresponds to the distance separating the principal image plane of the cornea and the principal object plane of the IOL.

^{6}Manufacturers label IOL powers with consistent values, but details of the IOL itself, such as geometrical and optical design, are not as readily available. For the same anatomical position, two implants of the same labeled power that are located at the same physical distance from the corneal endothelium will induce different refractive outcomes if their optical designs are not identical. This variation corresponds to differences in position of the principal object plane of any given IOL depending on its design. The lens constant compensates for these and many other parameters, such as minor systematic differences in power values that arise from the test methods.

^{6}

^{7}Such customization cannot be generalized and is only valid for a dedicated environment (e.g., for one surgical center with standardized surgical techniques and measurement equipment). Hence, a change of lens constant is generally necessary to optimize a formula to a different type of implant than that for which it has been originally adjusted. When more clinical data are available for various IOLs, their respective lens constants can be refined. The lens constant correlates the postoperative physical location of the lens component of the IOL to its effect on the final refraction.

^{8}

^{,}

^{9}In most formulas based on an optical model, a constant adjustment of the lens equates to modifying the predicted position of the IOL by a specific increment.

^{10}In this context, the modification of the constant is similar to an incremental change in the predicted position for all implants. In the case of a paraxial optical model in thick lenses, the modification of the lens constant is equivalent to the displacement of the principal object plane of the IOLs.

^{7}This trial-and-error constant optimization can also be performed for the eyes of a smaller external dataset of randomly chosen patients from a non-biased cohort.

^{11}

*ELP*) and predicted postoperative refraction for various IOL and corneal powers. A variation in

_{T}*ELP*may be induced by a variation in optical design and/or physical displacement of the implant. When the

_{T}*ELP*of each IOL is changed by the same increment, without any change in the design of its optics or its haptics, this is equivalent to an anatomical displacement of each IOL. We first investigated the theoretical variation in

_{T}*ELP*needed to compensate for a given systematic bias (non-null arithmetic mean of PE) as a function of the respective IOL and corneal powers of the considered eye. We then explored the relationships between incremental changes in

_{T}*ELP*and the SE prediction in spectacle plane corrections for various IOL and corneal powers. Finally, we calculated the difference in power applied to a given IOL to compensate for the change in spectacle refraction induced by a variation of its effective position.

_{T}^{12}Herein, we describe an explicit formula allowing back-calculation of the theoretical position of the principal object plane of an IOL. In what follows, the formulas are clearly reported for a pseudophakic eye modeled as thick lenses with four refractive surfaces and distinct refractive indices between the aqueous and vitreous humor (Fig. 1).

*n*). The total corneal power is denoted

_{s}*D*. It can be obtained from the value of the anterior and posterior radii of curvature (

_{c}*R*and

_{ca}*R*) and the refractive indices of air, stroma, and aqueous humor (

_{cp}*n*).

_{a}*D*. It depends on the curvatures of its anterior and posterior surfaces (

_{i}*R*and

_{ia}*R*), their separating distance (corresponding with the central IOL thickness,

_{ip}*d*), and the index variations between that of the media in contact with these surfaces and that of the lens itself. In this work, the power of the IOL was calculated from identical refractive indices of the aqueous humor and vitreous (

_{i}*n*= 1.336).

*ELP*) corresponds to the distance between the position of the principal image plane of the cornea and the principal object plane of the IOL. It can be computed from the corneal and IOL design characteristics and the anterior lens position (ALP), which separates the respective anterior surfaces of the cornea and the IOL. The axial length of the thick lens eye model (

_{T}*AL*) differs from the anatomical axial length (

_{T}*AL*). It is computed as the difference between the principal corneal image plane and the entire eye focal image point reduced by the distance between the principal planes of the IOL. However, the derived formulas apply to a thin lens eye model, where the axial length and the effective position of the implant have a more direct correlation.

_{A}*D*and IOL power

_{c}*D*, the distance separating the principal image plane of the cornea (

_{i}*H*′

*) from the focal image point of the entire eye (*

_{c}*F*′

*) is given by*

_{e}^{12}

*H*′

*=*

_{c}F_{e}*AL*, where

_{T}*AL*is the axial length of the thick lens model eye.

_{T}*ELP*Required to Compensate for a Systematic Bias

_{T}*R*) at the spectacle plane (

*d*) using the vergence formula:

*ELP*is equivalent to an anatomical displacement of the implant.

_{T}*ELP*allows us to compute the variation in the effective thick lens position (Δ

_{T}*ELP*) that is necessary to compensate for a change in spectacle refraction (Δ

_{T}*R*). Within the context of a zeroing procedure, this corresponds to a systematic bias (non-null mean PE) requiring compensation:

*ELP*on the Refraction of the Pseudophakic Eye

_{T}*ELP*on the refraction of the pseudophakic eye according to the power of the implant and the corneal diopter. This equation enables computation of the impact on

_{T}*R*of a given variation in

*ELP*(Δ

_{T}*ELP*), analogous to a change in the lens constant:

_{T}*D*> 0) between 1 D and 35 D.

_{i}*ELP*, corneal and IOL powers, and axial length.

_{T}*n*= 1.336 (aqueous),

_{a}*n*= 1.336 (vitreous),

_{v}*n*= 1.376 (corneal stroma), and

_{c}*n*= 1.45 (IOL material). All modeled IOL had a symmetrical shape (null Coddington shape factor). The central thickness of the cornea was

_{i}*t*=

_{c}*S*

_{1}

*S*

_{2}= 0.535 mm. The selected anterior and posterior corneal radii of curvature were

*R*= 8.7 mm and

_{ca}*R*= 7.5 mm (

_{cp}*D*= 38 diopters [D]);

_{c}*R*= 7.7 mm and

_{ca}*R*= 6.8 mm (

_{cp}*D*= 43 D); and

_{c}*R*= 6.9 mm and

_{ca}*R*= 6 mm (

_{cp}*D*= 48 D). The distance to the spectacle plane was set to

_{c}*d*= 12 mm.

*R*) Versus the Change in IOL Power (Δ

*D*) Induced by a Specified Change in ELP (Δ

_{i}*ELP*)

_{T}*D*be the power of an IOL located at the effective lens position (

_{i}*ELP*). The change in IOL power (Δ

_{T}*D*) required to keep the target refraction constant when an amount of Δ

_{i}*ELP*shifts the IOL position is given by

_{T}*H*′

*′′*

_{c}F*corresponds to the distance separating the principal image plane of the cornea from the focal point of the entire eye when the effective lens position is at*

_{e}*ELP*+ ∆

_{T}*ELP*(Fig. 2) and can be computed using Equation 1.

_{T}*R*) versus the required variation in IOL power (∆

*D*) induced by Δ

_{i}*ELP*for various combinations of corneal and IOL powers.

_{T}*ELP*between −0.17 mm and +0.17 mm by a ±0.1-mm increment and computed (1) the resulting change in refraction in the spectacle plane (Δ

_{T}*R*) and (2) the necessary change in IOL power (Δ

*D*) to maintain emmetropia. For each combination of IOL and corneal power, linear regression was performed between the corresponding values of Δ

_{i}*R*and Δ

*D*to obtain the regression coefficient.

_{i}*ELP*Required to Compensate for a Systematic Bias

_{T}*ELP*(Δ

_{T}*ELP*) to compensate for various systematic bias values (analogous to variations in refraction, Δ

_{T}*R*) from −0.3 D to +0.3 D by 0.1-D steps were computed for different IOL powers (

*D*) ranging from 1 D to 35 D (1-D steps) and three different total corneal powers (

_{i}*D*): 38 D, 43 D, and 48 D (Figs. 3a, 2b, 3c, respectively).

_{c}*ELP*increased dramatically with low-power IOLs (less than 10 D), proportional to the magnitude of the change in refraction (Δ

_{T}*R*). The flatter the cornea, the higher the difference required for the same planned refractive variation, but the incurred impact was low.

*ELP*on the Refraction of the Pseudophakic Eye

_{T}*R*) Versus the Change in IOL Power (∆

*D*) Induced by Δ

_{i}*ELP*

_{T}*ELP*). The regression coefficient expresses the change in spectacle refraction induced by a variation of 1 D in IOL power (Fig. 5). The regression coefficient values were between 0.617 and 0.731. The values of Δ

_{T}*R*and Δ

*D*obtained for Δ

_{i}*ELP*= +0.1 mm and Δ

_{T}*ELP*= −0.1 mm are indicated for each simulation. Table 2 displays the values for the 12 scenarios.

_{T}*R*), the required theoretical variation in

*ELP*is inversely proportional to the power of the implanted IOL. For implants with a power greater than 8 D, the theoretical variation in the effective lens position necessary to induce a refractive correction of ±0.1 D is less than 250 µm. For IOL powers less than 8 D, this variation increases exponentially: ±1 mm of a shift in

_{T}*ELP*is required to induce a refractive change of ±0.1 D for an IOL power of +2D. This tendency is relatively insensitive to the value of the corneal power.

_{T}*ELP*tends to be less for short eyes with high-power IOLs and increases dramatically for long eyes with low-power IOLs. This suggests that, for optimization methods that would be based on the determination of an average or median of the optimal constants calculated for each eye of a dataset, the determination of the lens constant will be greatly affected by the distribution of IOL powers within the dataset used. It also suggested that eyes with low-power IOLs will have a stronger influence than eyes with high-power IOLs.

_{T}^{12}This result suggests that the optimization processes induced by a change in implant design (without theoretical variation of their anatomical position) are subject to the same effects and the predominance of high-power implants. Due to manufacturing constraints and the variation in optical thickness related to changes in the refractive index and curvature of IOL surfaces, it is likely that the differences in paraxial refraction induced by different IOLs are related to the conjunction of anatomical displacement of the lens body and a variation in their optical design.

^{13}recently put forward convincing arguments in favor of choosing the standard deviation (SD) of the prediction error as the most relevant criterion for comparing the results of formulas with each other after lens constant adjustment is made to nullify the arithmetic mean PE. However, our results suggest that the lens constant value to cancel the systematic bias is likely to unpredictably vary the SD of the recalculated PE. The data presented in Figure 4 show the influence of the implant power on the variation of theoretical refraction for the same variation of the ELP. On the other hand, Table 2 reveals that the ratio between the variation of IOL power and the variation in refraction is relatively constant throughout the IOL power range. An example of an unfavorable scenario for optimization is a formula whose systematic bias would be mainly linked to a larger PE in long eyes (having received a low-power implant, such as <10 D). Our calculations show that the quality of the ELP prediction is not a major determining criterion for long eyes. Yet, as the zeroization process will eventually shift the predicted effective position for all eyes by a constant increment, it may degrade the performance of the formula on short eyes receiving high-power implants. Our results suggest that taking into account the power distribution of implants could be useful in the context of formula optimization.

^{14}

^{,}

^{15}It is often assumed that 1 D of IOL prediction error results in 0.7 D of refractive error at the spectacle plane.

^{16}

^{–}

^{18}However, the theoretical relationships between the refractive and the IOL power errors have not been extensively explored. Our results show that this ratio is valid for most biometric configurations but tends to be slightly different in extreme eyes. A constant increment of the IOL power induces a linear variation of the predicted refraction (Fig. 5), whose coefficient is relatively independent of the power of the considered IOL (Table 2). The compensation of a systematic refractive bias could be achieved by altering the target refraction by an amount equivalent to this bias. This would amount to adding or subtracting some constant value to the nominal power that would have been calculated for the IOLs without this adjustment. Such a zeroization process, based on a systematic variation added to the refractive target, would be less sensitive to the IOL power distribution of the dataset. It remains to be determined if the relative consistency of the ratio between the IOL and refractive prediction error would better preserve the performances obtained by the formula on the considered group before optimization when eliminating a systematic error. This would be achieved by adjusting for each eye (up or down) by an amount proportional to the arithmetic mean PE. Some of the dispersion and average error in refractive accuracy are related to variations in IOL design and anatomical position. These variations occur across powers of the same IOL type. Optimizing a formula by adding an offset to the target refraction would have a more consistent effect over the entire power range of the IOLs compared with altering the predicted ELP. However, a constant increment of the ELP is justified to address the impact of an average change in the expected position on postoperative refraction caused by a change in IOL style. To overcome these problems, it may be beneficial to know the design characteristics of the implants to improve the calculation methods using thick lens or ray-tracing-based IOL power formulas.

^{10}

^{,}

^{19}

^{,}

^{20}Instead of the straightforward calculation of the lens constants using formula inversion, nonlinear optimization algorithms have been developed with very high performance that could optimize any target parameter with any optimization criterion. It can be a measure that has high relevance for the patient and the patient's refractive outcome, such as the mean, the mean absolute, the median, or the root mean square error (RMSE) in terms of deviation of the achieved refraction after cataract surgery from the formula predicted refraction.

^{19}Some prerequisites and methods for successful formula constant optimization have been published recently

^{20}as a nonlinear gradient descent method to search for an optimized constant that yields the lowest mean absolute or RMSE.

*ELP*is inversely proportional to the IOL power of the considered eye for the compensated systematic bias. Subsequently, the variation of a lens constant has a greater influence on the power computation in short eyes. This is due to high-power IOLs having a significant impact on the predicted refraction, even for small variations in

_{T}*ELP*. The optimization of a formula based on multiple constants and nonlinear algorithms is certainly more robust against these biases. Nevertheless, any computational process requiring the adjunct of a constant shift in the predicted ELP value will be theoretically subject to these variations.

_{T}**D. Gatinel**, None;

**G. Debellemanière**, None;

**A. Saad**, None;

**R. Rampat**, None

*Diagnostics (Basel)*. 2022; 12(2): 243. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2008; 34(3): 368–376. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2008; 34: 1935–1939. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2017; 43(7): 869–870. [CrossRef] [PubMed]

*Acta Ophthalmol Scand*. 2007; 85(5): 472–485. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2021; 47(8): 1094–1095. [CrossRef] [PubMed]

*Am J Ophthalmol*. 2015; 160(3): 403–405. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2006; 32(3): 419–424. [CrossRef] [PubMed]

*PLoS One*. 2016; 11(7): e0158988. [CrossRef] [PubMed]

*Ophthalmic Res*. 2021; 64(6): 1055–1067. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2017; 43(8): 999–1002. [CrossRef] [PubMed]

*Transl Vis Sci Technol*. 2021; 10(4): 27. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2021; 47(1): 65–77. [CrossRef] [PubMed]

*Ophthalmology*. 2004; 111(10): 1825–1831. [CrossRef] [PubMed]

*PLoS One*. 2019; 14(11): e0224981. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2016; 57(9): OCT162–OCT168. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2013; 39(9): 1327–1335. [CrossRef] [PubMed]

*J Cataract Refract Surg*. 2019; 45(10): 1404–1415. [CrossRef] [PubMed]

*PLoS One*. 2021; 16(6): e0252102. [CrossRef] [PubMed]

*PLoS One*. 2022; 17(5): e0267352. [CrossRef] [PubMed]